Use implicit differentiation where $LaTeX: \displaystyle y$ is a function of $LaTeX: \displaystyle x$ to find the derivative of $LaTeX: \displaystyle 4 y^{3} e^{x} + 3 \log{\left(y \right)} \sin{\left(x^{3} \right)}=-3$

Taking the derivative of both sides using implicit differentiation gives: $LaTeX: 9 x^{2} \log{\left(y \right)} \cos{\left(x^{3} \right)} + 4 y^{3} e^{x} + 12 y^{2} y' e^{x} + \frac{3 y' \sin{\left(x^{3} \right)}}{y} = 0$ Solving for $LaTeX: \displaystyle y'$ gives $LaTeX: \displaystyle y' = - \frac{y \left(9 x^{2} \log{\left(y \right)} \cos{\left(x^{3} \right)} + 4 y^{3} e^{x}\right)}{12 y^{3} e^{x} + 3 \sin{\left(x^{3} \right)}}$